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Electromagnetic, Electrostatic and Magnetostatic Fields

Electromagnetic fields are characterized by coupled, dynamic (time- varying) electric and magnetic fields and are governed by the complete set of Maxwell’s equations (four coupled equations). According to Maxwell’s equations, a time-varying electric cannot exist without the a simultaneous , and vice versa. Under static conditions, the time-derivatives in Maxwell’s equations go to zero, and the set of four coupled equations reduce to two uncoupled pairs of equations. One pair of equations governs electrostatic fields while the other set governs magnetostatic fields. This decoupling of Maxwell’s equations illustrates that static electric fields can exist in the absence of magnetic fields and vice versa. Stationary charges produce electrostatic fields while magnetostatic fields are produced by steady (DC) currents or permanent .

Maxwell’s Equations (electromagnetic fields)

b ` Maxwell’s Equations Maxwell’s Equations (electrostatic fields) (magnetostatic fields) Electrostatic Fields

Electrostatic fields are static (time-invariant) electric fields produced by static (stationary) charges. The mathematical definition of the electrostatic field is derived from ’s law which defines the vector between two point charges.

Coulomb’s Law

Given point charges [q1, q2 (units=C)] in air located by vectors R1 and R2, respectively, the vector force acting on charge q2 due to q1 [F12 (units=N)] is defined by Coulomb’s law as

where is a unit vector pointing from q1 to q2 and åo is the free-space !12 [åo = 8.854×10 F/m]. The permittivity of air is approximately equal to that of free space (). Note that, according to Coulomb’s law, the force between the point charges is directly proportional to the product of the charges and inversely proportional to the square of the separation distance between the charges. The unit vector pointing from q1 to q2 can be written as

Inserting this equation for the unit vector into Coulomb’s law gives an alternative form of Coulomb’s law:

The first form of Coulomb’s law allows one to easily identify both the magnitude and direction of the vector force, while the second form does not require an explicit unit vector determination. Note that the unit vector direction is defined according to which charge is exerting the force and which charge is experiencing the force. This convention assures that the resulting vector force always points in the appropriate direction (opposite charges attract, like charges repel).

The point charge is a mathematical approximation to a very small volume charge. The definition of a point charge assumes a finite charge located at a point (zero volume). The point charge model is applicable to small charged particles (like ) or when two charged bodies are separated by such a large distance that these bodies appear as point charges to each other. Given multiple point charges in a region, the principle of superposition is applied to determine the overall vector force on a particular charge. The total vector force acting on the charge equals the vector sum of the individual . Force Due to Multiple Point Charges (Superposition)

Given a point charge q in the vicinity of a set of N point charges (q1, q2,..., qN), the total vector force on q is the vector sum of the individual forces due to the N point charges.

/ F total vector force on q due to q1, q2, ... , qN Electric Field

According to Coulomb’s law, the vector force between two point charges is directly proportional to the product of the two charges. Alternatively, we may view each point charge as producing a around it (electric field) which acts on any charge in its vicinity. Since a point charge will repel a charge of like sign, but attract a charge of unlike sign, we must adopt a convention as to the sign on the electric field force. We will adopt the convention that the direction of the vector electric field is the direction of the force on positive charge. Using a positive test charge to measure the electric field, the electric field is defined as the vector force per unit charge experienced by the test charge.

q - point charge producing the electric field

qt - positive test charge used to measure the electric field of q

RN- locates the source point PN (location of source charge q)

R - locates the field point P

(location of test charge qt)

From Coulomb’s law, the force on the test charge qt due to the charge q is

The vector electric field produced by q at the field point P (designated as E) is found by dividing the vector force on the test charge F by the test charge qt. Note that the electric field produced by q is independent of the magnitude of the test charge qt. The electric field units [Newtons per Coulomb (N/C)] are normally expressed as per meter (V/m) according to the following equivalent relationship:

For the special case of a point charge at the origin (RN = 0), the electric field reduces to the following spherical coordinate expression:

Note that the electric field points radially outward given a positive point charge at the origin and radially inward given a negative point charge at the origin. In either case, the electric field of the a point charge at the origin is spherically symmetric and the magnitude of the electric field varies as R!2.

The electric field due to multiple point charges can be determined using the principle of superposition. The vector force on a test charge qt at N N N R due to a system of point charges (q1, q2,..., qN) at (R1 , R2 ,..., RN ) is, by superposition,

On the spherical surface S of radius Ro, we have

Note the outward pointing normal requirement in Gauss’s law is a direct result of our electric field () convention.

By using an outward pointing normal, we obtain the correct sign on the enclosed charge. Gauss’s law can also be used to determine the electric fields produced by simple charge distributions that exhibit special . Examples of such charge distributions include uniformly charged spherical surfaces and volumes.

By symmetry, on S! (and S+), DR is uniform and has only an -component.

or

Gauss’s law can be applied on S+ to determine the electric field outside the charged sphere [E(R>Ro)].

or

Even integrand, Symmetric limits

(Potential in the x-y plane due to a uniform line charge of length 2a centered at the origin)

Electric Field as the of the Potential

The potential difference between two points in an electric field can be written as the line of the electric field such that

From the equation above, the incremental change in potential along the integral path is where è is the angle between the direction of the integral path and the electric field. The derivative of the potential with respect to position along the path may be written as

(a.)

0 0

(b.)